Mathematics

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20 pages, 10631 KiB

Open AccessArticle

Saliency Map Estimation Using a Pixel-Pairwise-Based Unsupervised Markov Random Field Model

by Max Mignotte

Mathematics 2023, 11(4), 986; https://doi.org/10.3390/math11040986 - 15 Feb 2023

Cited by 2 | Viewed by 1030

Abstract

This work presents a Bayesian statistical approach to the saliency map estimation problem. More specifically, we formalize the saliency map estimation issue in the fully automatic Markovian framework. The major and original contribution of the proposed Bayesian–Markov model resides in the exploitation of [...] Read more.

This work presents a Bayesian statistical approach to the saliency map estimation problem. More specifically, we formalize the saliency map estimation issue in the fully automatic Markovian framework. The major and original contribution of the proposed Bayesian–Markov model resides in the exploitation of a pixel pairwise modeling and a likelihood model based on a parametric mixture of two different class-conditional likelihood distributions whose parameters are adaptively and previously estimated for each image. This allows us to adapt our saliency estimation model to the specific characteristics of each image of the dataset and to provide a nearly parameter-free—hence dataset-independent—unsupervised saliency map estimation procedure. In our case, the parameters of the likelihood model are all estimated under the principles of the iterative conditional estimation framework. Once the estimation step is completed, the MPM (maximum posterior marginal) solution of the saliency map (which we show as particularly suitable for this type of estimation), is then estimated by a stochastic sampling scheme approximating the posterior distribution (whose parameters were previously estimated). This unsupervised data-driven Markovian framework overcomes the limitations of current ad hoc or supervised energy-based or Markovian models that often involve many parameters to adapt and that are finely tuned for each different benchmark database. Experimental results show that the proposed algorithm performs favorably against state-of-the-art methods and turns out to be particularly stable across a wide variety of benchmark datasets. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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29 pages, 2376 KiB

Open AccessArticle

Statistical Analysis of Inverse Weibull Constant-Stress Partially Accelerated Life Tests with Adaptive Progressively Type I Censored Data

by Mazen Nassar and Ahmed Elshahhat

Mathematics 2023, 11(2), 370; https://doi.org/10.3390/math11020370 - 10 Jan 2023

Cited by 7 | Viewed by 1287

Abstract

In life-testing investigations, accelerated life testing is crucial since it reduces both time and costs. In this study, constant-stress partially accelerated life tests using adaptive progressively Type I censored samples are taken into account. This is accomplished under the assumption that the lifespan [...] Read more.

In life-testing investigations, accelerated life testing is crucial since it reduces both time and costs. In this study, constant-stress partially accelerated life tests using adaptive progressively Type I censored samples are taken into account. This is accomplished under the assumption that the lifespan of products under normal use conditions follows the inverse Weibull distribution. In addition to using the maximum likelihood approach, the maximum product of the spacing procedure is utilized to obtain the point and interval estimates of the model parameters as well as the acceleration factor. Employing the premise of independent gamma priors, the Bayes point estimates using the squared error loss function and the Bayes credible intervals are obtained based on both the likelihood and product of spacing functions via the Markov chain Monte Carlo technique. To assess the effectiveness of the various approaches, a simulation study is used because it is not possible to compare the findings theoretically. To demonstrate the applicability of the various approaches, two real datasets for the lifetime of micro-droplets in the ambient environment and light-emitting diode failure data are investigated. Based on the numerical results, to estimate the parameters and acceleration factor of the inverse Weibull distribution based on the suggested scheme with constant-stress partially accelerated life tests, it is recommended to utilize the Bayesian estimation approach. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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17 pages, 961 KiB

Open AccessArticle

Advanced Approach for Distributions Parameters Learning in Bayesian Networks with Gaussian Mixture Models and Discriminative Models

by Irina Deeva, Anna Bubnova and Anna V. Kalyuzhnaya

Mathematics 2023, 11(2), 343; https://doi.org/10.3390/math11020343 - 09 Jan 2023

Cited by 4 | Viewed by 2146

Abstract

Bayesian networks are a powerful tool for modelling multivariate random variables. However, when applied in practice, for example, for industrial projects, problems arise because the existing learning and inference algorithms are not adapted to real data. This article discusses two learning and inference [...] Read more.

Bayesian networks are a powerful tool for modelling multivariate random variables. However, when applied in practice, for example, for industrial projects, problems arise because the existing learning and inference algorithms are not adapted to real data. This article discusses two learning and inference problems on mixed data in Bayesian networks—learning and inference at nodes of a Bayesian network that have non-Gaussian distributions and learning and inference for networks that require edges from continuous nodes to discrete ones. First, an approach based on the use of mixtures of Gaussian distributions is proposed to solve a problem when the joint normality assumption is not confirmed. Second, classification models are proposed to solve a problem with edges from continuous nodes to discrete nodes. Experiments have been run on both synthetic datasets and real-world data and have shown gains in modelling quality. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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19 pages, 512 KiB

Open AccessArticle

Bayesian and Non-Bayesian Inference for Weibull Inverted Exponential Model under Progressive First-Failure Censoring Data

by Abdullah Fathi, Al-Wageh A. Farghal and Ahmed A. Soliman

Mathematics 2022, 10(10), 1648; https://doi.org/10.3390/math10101648 - 12 May 2022

Cited by 4 | Viewed by 1589

Abstract

In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive first-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is verified. [...] Read more.

In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive first-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is verified. Via asymptotic normality of ML estimators and delta method, the corresponding confidence intervals (CIs) of the parameters and the reliability and hazard functions are constructed. For Bayesian inference, Lindley’s approximation and Markov chain Monte Carlo (MCMC) techniques are proposed to obain the Bayes estimators and the corresponding credible intervals (CRIs). To this end, both symmetric and asymmetric loss functions are used. A large number of Monte Carlo simulations are implemented to evaluate the efficiency of the developed methods. Eventually, a numerical example is analyzed for illustrative purposes. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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15 pages, 311 KiB

Open AccessArticle

A Bayesian EAP-Based Nonlinear Extension of Croon and Van Veldhoven’s Model for Analyzing Data from Micro–Macro Multilevel Designs

by Steffen Zitzmann, Julian F. Lohmann, Georg Krammer, Christoph Helm, Burak Aydin and Martin Hecht

Mathematics 2022, 10(5), 842; https://doi.org/10.3390/math10050842 - 07 Mar 2022

Cited by 5 | Viewed by 1928

Abstract

Croon and van Veldhoven discussed a model for analyzing micro–macro multilevel designs in which a variable measured at the upper level is predicted by an explanatory variable that is measured at the lower level. Additionally, the authors proposed an approach for estimating this [...] Read more.

Croon and van Veldhoven discussed a model for analyzing micro–macro multilevel designs in which a variable measured at the upper level is predicted by an explanatory variable that is measured at the lower level. Additionally, the authors proposed an approach for estimating this model. In their approach, estimation is carried out by running a regression analysis on Bayesian Expected a Posterior (EAP) estimates. In this article, we present an extension of this approach to interaction and quadratic effects of explanatory variables. Specifically, we define the Bayesian EAPs, discuss a way for estimating them, and we show how their estimates can be used to obtain the interaction and the quadratic effects. We present the results of a “proof of concept” via Monte Carlo simulation, which we conducted to validate our approach and to compare two resampling procedures for obtaining standard errors. Finally, we discuss limitations of our proposed extended Bayesian EAP-based approach. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

19 pages, 869 KiB

Open AccessEditor’s ChoiceArticle

Variational Bayesian Inference in High-Dimensional Linear Mixed Models

by Jieyi Yi and Niansheng Tang

Mathematics 2022, 10(3), 463; https://doi.org/10.3390/math10030463 - 31 Jan 2022

Cited by 5 | Viewed by 3083

Abstract

In high-dimensional regression models, the Bayesian lasso with the Gaussian spike and slab priors is widely adopted to select variables and estimate unknown parameters. However, it involves large matrix computations in a standard Gibbs sampler. To solve this issue, the Skinny Gibbs sampler [...] Read more.

In high-dimensional regression models, the Bayesian lasso with the Gaussian spike and slab priors is widely adopted to select variables and estimate unknown parameters. However, it involves large matrix computations in a standard Gibbs sampler. To solve this issue, the Skinny Gibbs sampler is employed to draw observations required for Bayesian variable selection. However, when the sample size is much smaller than the number of variables, the computation is rather time-consuming. As an alternative to the Skinny Gibbs sampler, we develop a variational Bayesian approach to simultaneously select variables and estimate parameters in high-dimensional linear mixed models under the Gaussian spike and slab priors of population-specific fixed-effects regression coefficients, which are reformulated as a mixture of a normal distribution and an exponential distribution. The coordinate ascent algorithm, which can be implemented efficiently, is proposed to optimize the evidence lower bound. The Bayes factor, which can be computed with the path sampling technique, is presented to compare two competing models in the variational Bayesian framework. Simulation studies are conducted to assess the performance of the proposed variational Bayesian method. An empirical example is analyzed by the proposed methodologies. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

22 pages, 5042 KiB

Open AccessArticle

A Novel Method to Use Coordinate Based Meta-Analysis to Determine a Prior Distribution for Voxelwise Bayesian Second-Level fMRI Analysis

by Hyemin Han

Mathematics 2022, 10(3), 356; https://doi.org/10.3390/math10030356 - 24 Jan 2022

Cited by 2 | Viewed by 2667

Abstract

Previous research showed that employing results from meta-analyses of relevant previous fMRI studies can improve the performance of voxelwise Bayesian second-level fMRI analysis. In this process, prior distributions for Bayesian analysis can be determined by information acquired from the meta-analyses. However, only image-based [...] Read more.

Previous research showed that employing results from meta-analyses of relevant previous fMRI studies can improve the performance of voxelwise Bayesian second-level fMRI analysis. In this process, prior distributions for Bayesian analysis can be determined by information acquired from the meta-analyses. However, only image-based meta-analysis, which is not widely accessible to fMRI researchers due to the lack of shared statistical images, was tested in the previous study, so the applicability of the prior determination method proposed by the previous study might be limited. In the present study, whether determining prior distributions based on coordinate-based meta-analysis, which is widely accessible to researchers, can also improve the performance of Bayesian analysis, was examined. Three different types of coordinate-based meta-analyses, BrainMap and Ginger ALE, and NeuroQuery, were tested as information sources for prior determination. Five different datasets addressing three task conditions, i.e., working memory, speech, and face processing, were analyzed via Bayesian analysis with a meta-analysis informed prior distribution, Bayesian analysis with a default Cauchy prior adjusted for multiple comparisons, and frequentist analysis with familywise error correction. The findings from the aforementioned analyses suggest that use of coordinate-based meta-analysis also significantly enhanced performance of Bayesian analysis as did image-based meta-analysis. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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12 pages, 393 KiB

Open AccessArticle

A First Approach to Closeness Distributions

by Jesus Cerquides

Mathematics 2021, 9(23), 3112; https://doi.org/10.3390/math9233112 - 02 Dec 2021

Viewed by 1368

Abstract

Probabilistic graphical models allow us to encode a large probability distribution as a composition of smaller ones. It is oftentimes the case that we are interested in incorporating in the model the idea that some of these smaller distributions are likely to be [...] Read more.

Probabilistic graphical models allow us to encode a large probability distribution as a composition of smaller ones. It is oftentimes the case that we are interested in incorporating in the model the idea that some of these smaller distributions are likely to be similar to one another. In this paper we provide an information geometric approach on how to incorporate this information and see that it allows us to reinterpret some already existing models. Our proposal relies on providing a formal definition of what it means to be close. We provide an example on how this definition can be actioned for multinomial distributions. We use the results on multinomial distributions to reinterpret two already existing hierarchical models in terms of closeness distributions. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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Review

Jump to: Research

51 pages, 2471 KiB

Open AccessReview

Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications

by Se Yoon Lee

Mathematics 2022, 10(6), 898; https://doi.org/10.3390/math10060898 - 11 Mar 2022

Cited by 10 | Viewed by 7024

Abstract

Nonlinear mixed effects models have become a standard platform for analysis when data is in the form of continuous and repeated measurements of subjects from a population of interest, while temporal profiles of subjects commonly follow a nonlinear tendency. While frequentist analysis of [...] Read more.

Nonlinear mixed effects models have become a standard platform for analysis when data is in the form of continuous and repeated measurements of subjects from a population of interest, while temporal profiles of subjects commonly follow a nonlinear tendency. While frequentist analysis of nonlinear mixed effects models has a long history, Bayesian analysis of the models has received comparatively little attention until the late 1980s, primarily due to the time-consuming nature of Bayesian computation. Since the early 1990s, Bayesian approaches for the models began to emerge to leverage rapid developments in computing power, and have recently received significant attention due to (1) superiority to quantify the uncertainty of parameter estimation; (2) utility to incorporate prior knowledge into the models; and (3) flexibility to match exactly the increasing complexity of scientific research arising from diverse industrial and academic fields. This review article presents an overview of modeling strategies to implement Bayesian approaches for the nonlinear mixed effects models, ranging from designing a scientific question out of real-life problems to practical computations. Full article

(This article belongs to the Special Issue Bayesian Inference and Modeling with Applications)

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